Wind-Tunnel Investigation of Aerodynamic Efficiency of Three Planar Elliptical Wings With Curvature of Quarter-Chord Line

Transcription

1 NASA Technical Paper 3359 October 1993 Wind-Tunnel Investigation of Aerodynamic Efficiency of Three Planar Elliptical Wings With Curvature of Quarter-Chord Line Raymond E. Mineck and Paul M. H. W. Vijgen

2 NASA Technical Paper Wind-Tunnel Investigation of Aerodynamic Efficiency of Three Planar Elliptical Wings With Curvature of Quarter-Chord Line Raymond E. Mineck Langley Research Center Hampton, Virginia Paul M. H. W. Vijgen High Technology Corporation Hampton, Virginia

3 Summary Three planar, untwisted wings with the same elliptical chord distribution but with dierent curvatures of the quarter-chord line were tested in the Langley 8-Foot Transonic Pressure Tunnel (8-Ft TPT) and the Langley 7- by 10-Foot High- Speed Tunnel ( HST). The dierent curvatures yielded a wing with an unswept quarter-chord line, a wing with an unswept trailing edge, and a wing with an unswept 150-percent chord (with a crescentshaped planform). A fourth wing with a rectangular planform and the same planform area and span as the elliptical-chord-distribution wings was also tested with two tip shapes. Force and moment measurements from the 8-Ft TPT tests are presented for Mach numbers from 0.3 to 0.5 and angles of attack from 04 to 7 for chord Reynolds numbers of 1: and 2: Sketches of the oil-ow patterns on the upper surfaces of the wings and some force and moment measurements from the HST tests are presented at a Mach number of 0.5. The aerodynamic eciency of the wings is evaluated by the lift-curve slope, the Oswald eciency factor, and a cambered-wing eciency factor. Fixing the boundary-layer transition near the leading edge of a wing that uses an airfoil designed for extensive laminar ow increases the drag coecient at zero lift and thickens the boundary layer. The thickened boundary layer decreases the eective camber of the airfoil which leads to a less-negative angle of zero lift. The reduction in the lift-curve slope and the reduction in the Oswald eciency factor with the xed transition indicate that the wing with the more extensive laminar boundary layer is more ecient. Increasing the Reynolds number decreases the drag coecient at zero lift. Increases in the lift-curve slope and the Oswald eciency factor with increasing Reynolds number indicate that the wing with the thinner boundary layer is more ecient. Increasing the curvature of the quarter-chord line makes the angle of zero lift more negative but has little eect on the drag coecient at zero lift. The changes in lift-curve slope and in the Oswald and cambered-wing eciency factors with the change in curvature of the quarter-chord line (wingtip location) indicate that the elliptical wing with the unswept quarter-chord line has the lowest lifting eciency and the elliptical wing with the unswept trailing edge has the highest lifting eciency; the crescent-shapedplanform wing has an eciency in between. Flow visualization results indicate that for lifting conditions, the ow near the tip on the upper surface of the elliptical wing with the unswept quarter-chord line is swept inboard. The ow near the tip of the elliptical wing with unswept trailing edge moves streamwise at the lower angles of attack and slightly outboard at the higher angle of attack. Flow near the tip of the crescent wing is swept outboard with a signicant separated ow region at the trailing edge of the tip. The ow at the tip of the elliptical wing with the straight trailing-edge and the ow at the tip of the crescent-shaped wing are probably inuenced by a vortex originating at the highly swept leading edges. Introduction Induced drag or drag due to lift constitutes approximately one-third of the total drag of conventional subsonic transport aircraft in cruising ight and as much as one-half of the total drag in climbing ight (ref. 1). For future aircraft with possible substantial amounts of laminar ow and reduced skinfriction drag, the relative contribution of induced drag will increase. In view of this possibility, methods to decrease induced drag for given total lift require renewed attention. Induced drag arises from rotational energy shed as vorticity into the wake of a nite-span lifting wing. The trailing vorticity induces a downwash that changes the local ow eld at the wing, and this results in a component of the total force on the wing in the drag direction. Approximating the wing by a bound vortex and assuming a planar, rigid wake (parallel to the free-stream direction), Munk showed in reference 2 that induced drag for a given lift is at a minimum when the downwash is constant across the span of the wing. An elliptical circulation distribution produces a constant downwash and, according to the classical linear theory, has minimum induced drag. Also based on the linear lifting-line assumptions, Munk's stagger theorem states that induced drag does not change if a specied total circulation is redistributed in the streamwise direction. Following the aforementioned assumptions of classical lifting-line theory, Cone showed in reference 3 that displacing the circulation distribution in the vertical direction theoretically oers large reductions in induced drag for a given total lift if an optimum circulation distribution is satised. The resulting concept of tip-mounted winglets, described in reference 4, has found application on some business jets and subsonic commercial transports. Reductions in drag due to lift by the addition of winglets can be as high as 10 to 20 percent, depending on the baseline conguration. (See ref. 5.) The potential induced drag benets of winglets may be oset by

4 increased structural complexity, increased weight, increased skin-friction drag from the increased wetted area, and adverse viscous and compressibility interference eects. Following Munk's stagger theorem, inplane curvature of the quarter-chord line does not aect the minimum induced drag once an elliptical chord distribution has been assumed to generate an elliptical loading. As exemplied in reference 6 (g. 11.4, p. 201), the denition of a planform shape with an elliptical chord distribution is not unique. Using linear theory, Burkett in reference 7 and Lowson in reference 8 indicated that a wing with a swept (or curved) quarter-chord line placed at an angle of attack shows a reduction in induced drag because of a vertical displacement of the outboard portion of the wing. Using the approach outlined by Cone, a maximum reduction in induced drag of 3 to 4 percent is predicted in reference 7 for an optimum circulation distribution and large angles of attack. To generate a given lift, a larger angle of attack is required for a wing with a symmetrical airfoil section than for a wing with a cambered airfoil section. For a wing with a swept (or curved) quarter-chord line, the larger angle of attack associated with the symmetrical airfoil section leads to a larger vertical displacement of the outboard portion of the wing. Thus, the generation of lift through angle of attack using an uncambered (symmetrical) airfoil section may have a small, but essential, induced-drag advantage over a similar wing using a cambered airfoil section. The vortex wake shed by a wing does not remain planar (rigid), as assumed by Munk, but deforms (relaxes) as it rolls up into the trailing wingtip vortices. Potential-ow computational methods have been developed that approximate the trailing-wake deformation. Drag predictions by surface-panel methods that allow an approximate relaxation of the trailing wake have indicated a reduction in induced drag for aft-swept and crescent-shaped elliptical planforms in comparison with those of the unswept elliptical wing (refs. 9 to 11). To the limit of discretization investigated, a reduction of 2 to 3 percent in induced drag is predicted for the crescent-shaped wing with an aspect ratio of 7 and a symmetrical airfoil (refs. 10 and 11). Wake relaxation reduced the Oswald eciency factor more for the unswept wing than for the crescent wing (ref. 11). The eciency computed for a given planform with the relaxed wake is less than the computed eciency with the rigid-wake assumption of Munk. From the inviscid calculations with wake relaxation, the spanloading near the tip of the crescent wing appears to be greater than the spanloading near the tip of the unswept elliptical wing (refs. 10 and 11). This increased loading near the tip can better approximate an elliptical loading. Obviously, a true elliptical load distribution is an unreachable goal since the vorticity shed at the tip approaches an in- nite value. Currently, work is underway to analyze the inviscid characteristics of curved elliptical planforms using Euler methods (refs. 12 and 13). Because the wake shape and location are obtained as an inherent part of the Euler solution, the need for approximating the shape of the trailing vortex wake in potential-ow methods using wake relaxation is eliminated. Available results from reference 13 indicate a 3-percent improvement in the lifting eciency for the crescent-shaped planform. Low-speed wind-tunnel experiments to investigate the eect of inplane modications on the drag due to lift have been reported in the literature (e.g., ref. 14). Tests of two untwisted wings with an aspect ratio (A) of 7 with the same elliptical chord distributions, one with an unswept quarter-chord line and the other with a crescent-shaped planform, indicated an improvement in the Oswald eciency factor of approximately 3 percent for the crescent planform at lift coecients below 0.5 (ref. 15). The streamwise airfoil shape was the uncambered NACA 0012 section, with boundary-layer transition xed near the leading edge. No signicant ow separation occurred for lift coecients below 0.5 for the two planforms. The maximum lift coecient of the crescent wing was 8 percent greater than that of the unswept wing (ref. 16). An analysis of the error propagation of this experiment in a large subsonic wind tunnel indicated that the absolute uncertainty in drag is the same order as the measured improvements because of the planform shape (ref. 15). The accuracy in the angle-of-attack measurement in these experiments appeared critical for accurate induced-drag measurements, a fact also noted by others (e.g., ref. 17). Classical, linear wing theory indicates that an elliptical span load distribution produces the minimum induced drag. Ignoring viscous eects and wake deformation, an untwisted wing with an elliptical spanwise variation of the chord has an elliptical span load distribution. Inviscid calculations with a relaxed wake indicate that increasing the curvature of the quarter-chord line will reduce the induced drag. Previous wind-tunnel experiments of two elliptical planform wings with dierent curvatures of the quarter-chord line (ref. 15) indicate that an induced-drag reduction occurs when the curvature of the quarter-chord line is increased. However, the uncertainty in those experimental measurements suggested that additional wind-tunnel experiments on similar wing shapes are needed with reduced 2

5 measurement uncertainty to determine if the induced-drag reduction predicted by the inviscid calculations occurs in a viscous ow. This report presents the results from experiments designed to study the induced drag of planar, elliptical planform wings with each having a dierent curvature of the quarter-chord line. The measurement uncertainty was reduced to allow the small dierences in the drag to be determined accurately. A cambered airfoil section was selected to provide a larger range of positive lift coecients without any signicant ow separation. An A = 6 wing was chosen to produce larger changes in induced drag compared with experimental results reported in reference 15. Four wings were tested in two wind tunnels. Three wings had the same elliptical, spanwise chord distributions but dierent curvatures of the quarterchord line. The fourth wing had a rectangular planform with the same area and span as the three elliptical wings. The purposes of the present investigation are as follows: (1) to study the lift and drag characteristics of elliptical chord wings with an increasing degree of curvature of the quarter-chord line and with a cambered streamwise airfoil section, (2) to determine the eect of transition location on the lift-dependent drag characteristics of the wings using a naturallaminar-ow airfoil, and (3) to assess the achievable accuracy and repeatability of induced-drag measurements using internal-force balances at medium subsonic speeds (at free-stream Mach numbers from 0.3 to 0.5) in a large transonic wind tunnel. The wings were rst tested in the Langley 8-Foot Transonic Pressure Tunnel (8-Ft TPT). Force and moment results are presented at Mach numbers of 0.3, 0.4, and 0.5 and angles of attack from about 04 to 7. The Reynolds numbers were 1: and 2: , based on the wing reference chord. The wings were subsequently tested in the Langley 7- by 10-Foot High-Speed Tunnel ( HST) to obtain surface ow visualization photographs and to obtain tunnel-to-tunnel repeatability data. The Reynolds number varied from 1: to 1: as the Mach number varied from 0.3 to 0.5 because the 7210 HST is an atmospheric tunnel. The ow visualization tests covered the same Mach number and angle-of-attack ranges. Symbols The longitudinal aerodynamic characteristics are presented in the stability-axes system. Results are presented in coecient form with the model moment reference center at the quarter-chord location of the wing root. All measurements and calculations were made in the U.S. customary units, and dimensional results are presented in the U.S. customary units. A b C D C D;min C 3 D;min C D;0 C L C L;min wing aspect ratio, b2 S wing reference span (48.00 in.) Drag drag coecient, q1s minimum drag coecient minimum prole drag coecient drag coecient at zero lift lift coecient, Lift q1s lift coecient at minimum drag coecient C 3 L;min lift coecient at minimum prole drag coecient C L lift-curve slope, deg 01 Cm c croot c D e e 3 K i Kp L l M1 q1 Rc Rs R pitching-moment coecient about wing-root quarter-chord location, Moment q1sc local chord, in. wing chord at model centerline, in. wing reference chord (8.00 in.), S b drag, lb Oswald eciency factor, A dc 01 D dc 2 L cambered-wing eciency factor, A dcd 01 d(cl0cl;min) 2 inviscid induced-drag factor (see eq. (8)) viscous induced-drag factor (see eq. (8)) lift, lb length of nose section (6.00 in.) free-stream Mach number free-stream dynamic pressure, psi free-stream Reynolds number based on model reference chord free-stream Reynolds number based on chordwise distance along surface local Reynolds number based on boundary-layer momentum thickness 3

6 r l r max S x x le x tip y z 0 Abbreviations: diam. rms Sta. Wind Tunnels local body radius, in. maximum body radius (1.50 in.) wing planform reference area ( in 2 ) streamwise distance from wing-root leading edge, in. streamwise distance from wing-root leading edge to local leading edge, in. streamwise distance from wing-root leading edge to wingtip, in. spanwise position, in. normal position, in. geometric angle of attack, deg angle of attack at zero lift, deg position on semispan, diameter root mean square station y b=2 The Langley 8-Foot Transonic Pressure Tunnel The majority of the experiments were conducted in the Langley 8-Foot Transonic Pressure Tunnel (8-Ft TPT). Information about this tunnel may be found in reference 18. The tunnel is a singlereturn, fan-driven, continuous-operation pressure tunnel. The test section is 160 in. long with an 85.5-in-square cross section and corner llets. The top and bottom walls have four longitudinal slots yielding a porosity of about 5 percent, and the sidewalls are solid. The empty test section Mach number is continuously variable from about 0.20 to Stagnation pressure can be varied from 0.25 to 2.00 atm. Air dryers are used to control the dew point. A heat exchanger located upstream of the settling chamber controls the stagnation temperature. The test section contraction ratio is 20.25:1, and there are ve turbulence-reduction screens. An arc-sector model support system is located in the high-speed diuser. The angle range of the arc sector is from 012:5 to 12:5. The whole arc sector can be translated longitudinally to position the model at the desired test section station for testing. The Langley 7- by 10-Foot High-Speed Tunnel Flow visualization and tunnel-to-tunnel data repeatability studies were performed in the Langley 7- by 10-Foot High Speed Tunnel ( HST). A general description of the tunnel and its support equipment is found in reference 19. It is a single-return, closed-circuit, fan-driven wind tunnel, and it operates at ambient temperature and pressure. Test section walls are solid with no divergence. Streamwise fairings on the sidewalls modify the cross-sectional area distribution to provide a uniform longitudinal Mach number distribution along the centerline of the test section. The test section is 6.58 ft high by 9.57 ft wide with a useable length of ft. A variable-speed drive motor provides a continuous Mach number range from near 0 to The contraction ratio is 17:1 and there are four turbulence-reduction screens. The model support system consists of a vertical strut and a variablepitch-angle sting support system with a range from 012 to 12. The sting support system can be translated on the vertical strut to position the model close to the test section centerline. Models Four wing-body models were tested. Each model consisted of a common forebody, a wing with an integral centerbody, and a common aftbody. One wing had a rectangular planform and an 8.00-in. chord. The other three wings had the same elliptical spanwise variation of chord. The physical characteristics of the wings are listed in table 1. Photographs of one of the models installed in the 8-Ft TPT are presented in gure 1. A sketch of the model with one of the wings is presented in gure 2(a). All elliptical planform wings had a span of in. and a projected area of in 2, which yielded a common aspect ratio of Each wing was untwisted. The NASA NLF(1)-0416 airfoil section was used for all four wings. This cambered airfoil is 16 percent thick and is designed for a lift coecient of 0.4 at a Reynolds number of Details of the airfoil characteristics are presented in reference 20, and the airfoil coordinates are listed in table 2. A sketch of the airfoil section is found in gure 2(b). The airfoil chord coincided with the centerline of the body. The quarter-chord location of the wing root was located in. aft of the beginning of the forebody for all four wings. The model forebody was 6.30 in. long with the rearmost 0.30 in. at a constant diameter of 3.00 in. 4

7 The forward 6.00 in. was dened by the polynomial r l x x 2 x 3 1=2 = (1) r max l l l A sketch of the model forebody is presented in gure 2(c), and the coordinates are listed in table 3. The integral centerbody was in. long with a diameter of 3.00 in. The model aftbody was a straight cylinder with a diameter of 3.00 in. and a length of 9.00 in. The internal diameter at the downstream end of the aftbody was 2.90 in. The three elliptical wings have the same spanwise variation of chord but dierent planforms because of dierent curvatures of the quarter-chord line. The chord at the model centerline on each of the elliptical wings was in. The spanwise distribution of the local chord was elliptical and was determined from the chord at the model centerline (c root ) and the model span (b). Thus, c = c root s1 0 y b=2 2 = crootq1 0 2 (2) The planform view of the leading-edge shape was also chosen to be elliptical: 2 s 3 2 x le = xtip = xtip 1 0 p (3) y b=2 The streamwise position of a point relative to the wing-root leading edge (x) on the wing at a specied fraction of the local chord ((x=c) local ) is obtained from equations (2) and (3) as follows: x x()=x le ()+ c() c local 1 0 p x + crootp = x tip = c root nh x c local 0 x tip c root c local i p x tip c root o (4) The streamwise position of a point (x()) at a constant nondimensional chordwise position ((x=c) local ) at a spanwise station () is determined by the nondimensional location of the wingtip (x tip =c root ). Therefore, assuming the above expressions for the elliptical distributions for the chord and for the leading edge, the elliptical wing planform shape is determined by the nondimensional location of the wingtip. Equation (4) can be used to determine the line de- ned by a constant fraction of the local chord, such as the quarter-chord line. Note that x() is a constant (equal to x tip ) if the selected fraction of the local chord ((x=c) local ) is equal to the nondimensional location of the wingtip (x tip =c root ). For this case, the curve dening the constant fraction of the local chord is a straight line in the spanwise direction passing through the tip. The three nondimensional locations selected for the wingtip (x tip =c root ) for this study were 0.25, 1.00, and Sketches of the three elliptical wings are presented in gure 3. The wing with the nondimensional wingtip location of 0.25 has an unswept quarter-chord line and will be referred to as wing A. The wing with the nondimensional wingtip location of 1.00 has an unswept trailing edge and will be referred to as wing B. The wing with the nondimensional wingtip location of 1.50 has a crescentshaped planform and will be referred to as wing C. The curvature of the quarter-chord line increases as the nondimensional tip location increases from 0.25 to A fourth wing was tested as a baseline planform. This wing, referred to as wing D, had a rectangular planform. A comparison of wings A and D is presented in gure 4. Wing D had two sets of interchangeable tips. The wing with the square tip had the same span, area, and aspect ratio as the elliptical planform wings. The wing with the round tip had a span of in. and a projected area of in 2. The resulting aspect ratio was The rounded end of the tip was formed by revolving one-half of the airfoil section local thickness about the camber line. A sketch of the round tip superimposed on the square tip is presented in gure 5. Photographs of the rectangular wing with the square tips and with the rounded tips are shown in gure 6. A brief summary of the four wings is presented in chart A. Chart A Nondimensional wingtip location, Wing x tip =c root Description A 0.25 Unswept quarter-chord line (elliptical chord distribution) B 1.00 Unswept trailing edge (elliptical chord distribution) C 1.50 Crescent-shaped planform (elliptical chord distribution) D N/A Rectangular planform (round tip and square tip) 5

8 Instrumentation The models were mounted on a six-component strain gauge balance supported by a straight sting as shown in gure 7. The sting was mounted directly to the model support system without any knuckles so that the model support system and the sting were aligned. This sting was used in both wind-tunnel tests. Dierent strain gauge balances were selected for each tunnel test to match the maximum airload on the model with the balance maximum load capacity. The measurement accuracy for each component of the strain gauge balance was determined from the calibration of the balance for the load ranges encountered in the test. For the 8-Ft TPT test, the measurement accuracies for the components of the strain gauge balance were 60.1 percent of the full-scale load for normal force and pitching moment and less than 60.3 percent of the full-scale load for axial force. These accuracies correspond to lbf for normal force, lbf for axial force, and in-lbf for pitching moment. To measure the model angle of attack, an accelerometer was installed inside the nose of the forebody and attached to the front surface of the balance mounting block that was common to all models. The static calibration of the accelerometer was accurate to within 60:01 over the range of angles of attack presented in this report. Two tubes were installed on the left and right sides of the sting extending into the aftbody to measure the chamber pressure within the model for use in computing the correction to axial force (and drag) to a condition of free-stream static pressure at the base of the model. Free-stream total and static pressures were measured with sonar mercury manometers. The accuracy of the sonar manometers was 60.3 psf. Uncertainties (U) in the Mach number, lift coecient, and drag coecient for the test in the 8-Ft TPT are derived in the appendix. The uncertainties in the Mach number were computed at nominal values of the free-stream total and static pressures. The Mach number uncertainty was typically about The uncertainties in the lift and drag coecients were determined for measured model loads over a range of angles of attack from 03 to 3 because results over this range will be used to compare the eciencies of the dierent wings. As expected, the uncertainties were largest at the lowest Mach number and Reynolds number where the air loads were smallest. The uncertainty became smaller as the Mach number and/or Reynolds number increased. For these small angles of attack, the accuracy of the normal-force measurement had the strongest inuence on the uncertainty of the lift coecient. The accuracy of the axial force and, to a lesser degree, the accuracy of the angle-of-attack measurement had the strongest inuence on the uncertainty of the drag coecient. At a Mach number of 0.5, the uncertainty in the lift coecient was about and the uncertainty in the drag coecient was about Boundary-Layer Transition Strips Changes in the location of the boundary-layer transition from laminar to turbulent ow will change the boundary-layer thickness and the viscous drag. The chordwise location of transition depends on the Reynolds number and the pressure distribution, which is a function of the angle of attack. The transition location also depends on the leading-edge sweep angle because of the crossow and attachment-line instabilities, and hence the location will probably be dierent for each elliptical-planform wing model. If the transition location changes with lift, the viscous drag and total drag will also change. Also, at the low Reynolds numbers near the tip of the elliptical planform wings, nonreattaching laminarseparation bubbles may occur. These eects complicate the analysis of the results. To minimize these eects, the location of the boundary-layer transition was xed by applying strips of carborundum grit to the model surface. For all tests, a ring of transition grit (0.06 in. wide) was placed on the forebody at a location 0.75 in. back along the surface. For the transition-xed tests, strips of transition grit were placed on both the upper and lower surfaces of the wing, as shown in gure 8. A constant chordwise location was selected because that is a requirement of many Computational Fluid Dynamics (CFD) codes. The 0.075c position was selected for the transition strips on the wing. The grit size was determined by using the essentially twodimensional methods of Braslow and Knox in reference 21 for R c =2: ;M1 =0:5, and a freestream Reynolds number of 600, based on a reference length equal to the grit height. Sketches showing this baseline grit installation, used on wings A and C, are found in the upper part of gures 8(a) and 8(c). In the tip region of the elliptical wings, the most forward transitiongrit location that does not violate the criterion that R s > 0: is aft of the 0:075c location selected. Large grit sizes, with associated nonnegligible grit drag and changes in local ow eld, are necessary to promote transition for R s << 0: (refs. 22 and 23). The baseline grit installation maintained the 0:075c trip location all the way out to the wingtip. Using the sublimating-chemical, transition 6

9 visualization technique, the eectiveness of the trip was studied on wing A at = 00:25 ;M1 =0:3, and R c =2: The trip strip was eective at tripping the boundary layer across the span of the wing except for the tip region. Uncertainty about the effectiveness of the trip near the wingtip led to a modication of the grit size and location in the outboard region in later phases of the experiment. Sketches showing the modied grit installation are found in gure 8. For wings A and B, from =0:96 to the tip, grit was installed in a straight line. For wing C, the grit was installed in a straight line from =0:94 to the tip because of the larger local leading-edge sweep angle. Larger values of R s are realized by the modication, but the chordwise grit location is no longer at a constant fraction of the local chord. A comparison of the lift and drag characteristics of wing A and wing C for the two trip locations, presented in gures 9 and 10, respectively, shows no noticeable dierences in the lift and pitching-moment coecients. The drag coecient at a given lift coef- cient for the modied grit location is slightly less than that for the baseline grit location. Variations in spanwise and chordwise locations of transition among the wings due to the increasing degree of leading-edge sweep can result in variations in viscous drag among the wings as the angle of attack is varied. Attachment-line transition as well as crossow instability can occur in regions where the leading-edge sweep is suciently large. These possible viscous drag variations among the wings can be improperly interpreted as dierences in inviscid lift-dependent drag. In an attempt to further reduce the variation in transition location in the leadingedge region, a trip strip was installed in a direction normal to the leading edge on wings B and C. The strip extended back to the constant-chord grit strips on both surfaces. A similar trip, which was also installed in the experiments made by Van Dam, Vijgen, and Holmes, was reported in reference 15. The spanwise location of this trip was chosen to coincide with a predicted attachment line at R = 100. If R exceeds 100, the boundary layer along the attachment line will remain turbulent when a large trip is present (ref. 24). Using the dimensional leading-edge radius in the normal direction and the leading-edge sweep angle as a function of spanwise location, R was estimated for a given free-stream Reynolds number by assuming an innite swept-wing geometry and the stagnation line on the leading edge (ref. 25). Only wings B and C had sucient leading-edge sweep to yield a value of R of 100 at the Reynolds numbers used in this test; therefore, wing A did not have a leading-edge trip. The leading-edge trip for wing B was located at =0:90, as shown in gure 8(b), and the leading-edge trip for wing C was located at = 0:75, as shown in gure 8(c). Tests and Procedures The model was tested in the Langley 8-Foot Transonic Pressure Tunnel at Reynolds numbers of 1: and 2: , based on the wing reference chord. Tests at the lower Reynolds number were conducted at M1 =0:3, and tests at the higher Reynolds number were conducted at M1 = 0:3;0:4, and 0.5. The angle of attack was varied from 04 up to 7. At an angle of attack of 0, the nose of the model was located on the test section centerline 80 in. downstream from the start of the test section so that the model was located approximately in the middle of the useable portion of the test section. Because the center of rotation of the model support system was near the base of the sting, the model location moved slightly downstream and upward as the angle of attack was increased from 0. To minimize any aerodynamic hysteresis eects, all angles of attack were approached from below the desired angle. All four wings were tested with xed transition. Only wing A was tested with free transition. Tests of wings A and C in an upright and inverted orientation led to an average downwash of 0:037 for this model location and these planforms. The angularity was constant for the ranges of angles of attack and Mach numbers considered herein. This correction was applied to all the results from the 8-Ft TPT. The two model chamber pressures were averaged and used to correct the balance axial force (and drag) for the pressure at the open end of the aftbody. No corrections were applied for model blockage or jet boundary eects. The model was subsequently tested in the HST on the same sting that was used for the tests in the 8-Ft TPT. A smaller capacity balance was used for these tests because the dynamic pressure and, consequently, the airloads were reduced in the HST. The models were tested at M1 = 0:3;0:4, and 0.5 over the same angle-ofattack range. The Reynolds number varied from 1: to 1: as the Mach number varied from 0.3 to 0.5. The same test procedures were used to minimize aerodynamic hysteresis effects. Surface oil-ow visualization photographs were taken at several angles of attack on each wing at Mach numbers of 0.3 and 0.5. All tests in the HST were conducted with the modied grit congurations. 7

10 Tests of wings A and D in an upright and inverted orientation led to an average downwash for these planforms at this model location that varied slightly with Mach number. The ow-angularity corrections at M1 =0:3;0:4, and 0.5 were 0:025,0:040, and 0:040, respectively. This correction was applied to all the results from the HST. The model chamber pressures were averaged and used to correct the balance axial force (and drag) for the pressure at the open end of the aftbody. Corrections were applied for model blockage and jet boundary eects using the techniques of references 26 and 27. Overall wing eciency was determined by examining three parameters: the lift-curve slope 0 C L 1, the Oswald eciency factor (e), and the camberedwing eciency factor (e 3 ). The changes in these characteristics were expected to be small. Least-squares curves were tted to the measured data to determine the slopes of each curve in a consistent manner. To determine the lift-curve slope, the typical linear variation of lift coecient with angle of attack was assumed: C L = C L ( 0 0 ) (5) Inspection of the lift curves with xed transition indicated that they were fairly linear over an angleof-attack range from 03:1 to 3:1. A linear leastsquares curve was t to the results within this angleof-attack range to determine the angle of zero lift ( 0 ) and the lift-curve slope (C L ). To determine the Oswald eciency factor, the drag was assumed to vary linearly with the square of the lift coecient: C D = C D;0 + 1 Ae C2 L (6) Inspection of the curves showing the variation of C D with CL 2 indicated that the curves became nonlinear at low lift coecients because of the airfoil camber drag. The curves also were nonlinear at lift coecients above about 0.5, which is near the maximum value of L=D (not presented). A linear least-squares curve was t to the results for lift coecients between 0.15 and 0.50 to obtain the slope (dc D =dc 2 L ). The lower limit was selected to eliminate the nonlinear points near zero lift, and the upper limit was selected to eliminate the nonlinear points at the higher angles of attack where trailing-edge separation occurred. The Oswald eciency factor (e) was computed from this slope. Because of the linear curve, the slope of the curve and the value of the Oswald eciency factor are constant. Because the wing utilized a cambered airfoil section, the minimum drag did not occur at zero lift but at a small positive lift. Thus, equation (6) is not the best representation of the variation of drag coecient with lift coecient for a wing with camber. Inspection of the variation of C D with C L (not shown) revealed the expected parabolic variation centered about C L;min, that is, the lift coecient associated with the minimum total drag coecient (C D;min ). A more representative expression for the results is C D = C D;min Ae 3 CL 0 2 C L;min (7) This equation was used to determine the camberedwing eciency factor (e 3 ). A quadratic least-squares curve was tted to the drag data for lift coecients below 0.50, and the results were used to determine the minimum total drag (C D;min ), the associated lift (C L;min ), and the drag at zero lift (C D;0 ). Using the computed value of C L;min, a linear least-squares curve was tted to the variation of C D with (C L 0 C L;min ) 2 for lift coecients up to 0.50 to obtain the slope dcd d(cl0cl;min )2. This slope was used to determine the cambered-wing eciency factor (e 3 ). Results from the curve ts, presented in table 4, will be discussed with the appropriate data. The viscous drag is generally a weak function of the angle of attack and, hence, of the lift coecient. As shown in reference 14, a viscous contribution occurs to the drag due to lift (the K p term) in addition to the inviscid contribution (the K i term): C D = C 3 D;min + K p(c L 0 C 3 L;min )2 + K i C 2 L A (8) As outlined in reference 15, the choice of an airfoil shape that has little or no variation of airfoil drag with lift squared (K p 0, at least for the lower angles of attack) allows a reduction of the viscous drag contribution to the lift-dependent drag variation. As a consequence, better agreement between the inverse of the theoretical inviscid induced-drag factor (1=K i ) and the experimental Oswald eciency factor (e) can be expected. As noted above, equation (7) is a better representation of the variation of the drag coecient with lift coecient for a wing with a cambered 8

11 airfoil section than equation (6). By using equation (7) to compute dc D =dcl 2 and the denition for the Oswald eciency factor given in the section dening the symbols, the following relationship of the Oswald eciency factor to the cambered-wing eciency factor is derived: e = e3 1 0 (C L;min =C L ) (9) This equation indicates that for positive values of C L;min, the Oswald eciency factor is greater than the cambered-wing eciency factor. Presentation of Results The aerodynamic data presented in this report are identied by a unique \run" number. Results are presented in coecient form in the stability-axes system. The same reference area was used for all four wings, although the projected area for wing D with the round tips was slightly larger than the area of the other wings. The same wing reference chord was used to nondimensionalize all pitching-moment data. The common aspect ratio of the elliptical wings was used to determine the Oswald eciency factor and the cambered-wing eciency factor. All results are with the transition xed unless specically noted otherwise. Aerodynamic characteristics presented in the following gures were measured in the 8-Ft TPT unless otherwise noted. Flow visualization results presented herein were derived from photographs taken in the HST. Parameters derived from the curve ts of the results are listed in table 4. The drag coecient at zero lift was derived from the quadratic curve t because it represented the data better than the linear curve t. The results are presented as follows: Figure Aerodynamic characteristics from the 8-Ft TPT: Data repeatability for wing A Eect of transition for wing A Eect of Reynolds number for wing A , 14 Eect of Mach number at R c =2: for Wing A Wing B Wing C Wing D Eect of tip shape for wing D Eect of elliptical wing planform Comparison of aerodynamic characteristics for wing A from the 8-Ft TPT and the HST Surface ow visualization sketches from the HST: Wing A Wing B Wing C Wing D Wing eciency: Eect of xing transition on wing eciency parameters: Variation of C L with M Variation of e with M Variation of e 3 with M Eect of wing planform on wing eciency parameters: Variation of C L with R c Variation of e with R c Variation of e 3 with R c Variation of C L with M Variation of e with M Variation of e 3 with M

12 Discussion of Results This investigation centered on the eects of different parameters on wing eciency as quantied by the lift-curve slope (CL), the Oswald eciency factor (e), and the cambered-wing eciency factor (e 3 ). Repeatability of the basic aerodynamic characteristics is discussed rst. The eects of boundarylayer transition, Reynolds number, Mach number, and wing planform on the basic aerodynamic characteristics are then presented. Results from the owvisualization tests are discussed next. Finally, the effects of boundary-layer transition and wing planform on the wing eciency parameters are discussed. Data Repeatability Wings A and C were tested at each Mach number and Reynolds number more than once. The repeatability of the results is excellent. A sample of the lift, drag, and pitching-moment data from wing A is presented in gure 11 for a Mach number of 0.3. The results from four runs are represented quite well by a single, representative curve. Note that the data include both model upright and inverted results. The lift curve was linear over the angle-of-attack range from 03:1 to 3:1. Linear least-squares curves were tted to the lift data within this angle-of-attack range for each run. The average deviation of the measured lift coecient from the curve t was , and the standard deviation was ; this was less than the uncertainty in the lift coecient of The dierence between the maximum and minimum angles of zero lift was only 0:02, which was slightly larger than the accuracy of the angle-of-attack measurement of 0:01. The lift-curve slopes ranged in value from per degree to per degree. The drag coecient varied linearly with the lift coecient squared over the range from 0.02 to (The lift coecient ranged from 0.15 to 0.50.) Linear least-squares curves were tted to the drag data over this range of lift coecient squared. The standard deviation of the measured drag data from the curve t was about , which was much smaller than the computed uncertainty in drag of As was found with the lift results, the dierent runs were in very good agreement with each other. The dierence between the maximum and minimum values of the drag coecients at zero lift was , which was about the same as the uncertainty. The slopes of these curves were also in good agreement with each other. The pitching-moment curves were very repeatable. The high quality of the results allowed small dierences to be discerned in the lift and drag, thus making them suitable for studying the eects of planform on the wing eciency parameters. Eect of Fixing the Location of Boundary-Layer Transition Wing A was tested with free transition and xed transition on the wing to determine the aerodynamic characteristics with and without a long run of a laminar boundary layer. The results, presented in gure 12 for a Mach number of 0.5, are typical of those measured with free and xed transition at all three Mach numbers. Results from curve ts of the data at the other Mach numbers can be found in table 4. The thicker turbulent boundary layer with xed transition tends to decrease the eective camber of the airfoil more than the thinner laminar boundary layer with free transition. Thus, the angle of zero lift is about 0:33 less negative and the lift-curve slope is reduced by per degree for xed transition. With free transition, the location of boundary-layer transition changes with angle of attack. Evidence of this change is the small nonlinearity in the lift curves with free transition that is apparent by close visual inspection of the plotted curves at negative angles of attack and by the increased rootmean-square (rms) error obtained from the linear curve ts. The slopes of the lift curves with free transition become smaller at small positive angles of attack. The change in slope may be associated with changes in the location of the boundary-layer transition on the upper surface and the presence of laminar separation bubbles. Fixing transition eliminates the long run of the laminar boundary layer and should reduce the possibility of laminar separation bubbles. With xed transition, the lift curves are more nearly linear between 03 and 3. The higher skin friction of the turbulent boundary layer leads to an increase in the drag coecient at zero lift of about This increase in drag coecient is similar to the measured increase in minimum airfoil section drag coecient for the NASA NLF(1)-0416 airfoil section (ref. 20). The pitching moment with xed transition is less negative because the thicker turbulent boundary layer with xed transition decreases the eective camber of the airfoil more than the thinner boundary layer with free transition. The slope of the pitchingmoment curve with xed transition is constant over most of the angle-of-attack range. At positive angles of attack, the slopes with free and xed transition are similar. At these conditions for free transition, the location of transition on the upper surface may have moved upstream to match the xed-transition location. These results are consistent with the change in lift-curve slope at small positive angles of attack. 10

13 Eect of Reynolds Number Boundary-layer growth (which is dependent on the Reynolds number) and the associated viscous drag variation inuence the wing eciency. Each wing was tested at two Reynolds numbers for a Mach number of 0.3. Results are presented for wing A in gure 13 with free transition and in gure 14 with xed transition. In general, increasing the Reynolds number decreases the boundary-layer thickness at a given location. The lift-curve slope is higher for the higher Reynolds number (with a thinner boundary layer). Increasing the Reynolds number leads to a decrease in the drag coecient at zero lift and a decrease in the slope of the drag coecient versus the square of the lift coecient curves (an increase in the Oswald eciency factor). The pitching moment becomes more negative at the higher Reynolds number because the thinner boundary layer does not reduce the eective camber as much as a thicker boundary layer. These trends in the lift-curve slope and Oswald eciency factor going from the lower to the higher Reynolds number (with a decreasing boundary-layer thickness) are similar to the trends going from xed transition to free transition (with a decreasing boundary-layer thickness). The eect of Reynolds number on the results from wings B, C, and D with transition xed (which is not presented herein) is similar to the eect of Reynolds number on the results from wing A with transition xed. The results from curve ts of the data from wings B, C, and D may be found in table 4. Eect of Mach Number Tests on each wing were conducted at a chord Reynolds number of 2: for three Mach numbers: 0.3, 0.4, and 0.5. At these Mach numbers and moderate lift coecients, the ow should be subcritical everywhere on the wing. The eect of Mach number on the aerodynamic characteristics of each wing is presented in gures 15 to 18. Changes with Mach number are similar for all four wings. In general, for a given wing, the angle of zero lift becomes less negative and the lift-curve slope increases as the Mach number increases. Increasing the Mach number increases the drag coecient at zero lift slightly and makes the pitching-moment coecient more negative. Eect of the Rectangular Wingtip Shape Wing D, which has a rectangular planform, was tested both with tips that had square ends and with tips that had round ends. All data have been reduced by the wing reference area ( in 2 ). The projected area of wing D with the round tips is in 2. The aerodynamic coecients for wing D with the round tip can be renormalized by the projected area by multiplying by A comparison of the results with the dierent tips is presented in gure 19. In spite of the increase in area of the round tip, the tip shape has little apparent eect on the lift. Results from the curve ts indicate that changing the tip shape has no measurable eect on the angle of zero lift. The lift-curve slope is slightly smaller for the wing with the round tip although it has a larger actual planform area. Renormalizing the round tip results with the actual planform area would further reduce the liftcurve slope for the round tip. Thus, for a given angle of attack, the loading on the wing with the round tip is less than the loading with the square tip. The drag coecient at zero lift is larger for the wing with the square tip, possibly because of the separated ow over the face of the square tip. The slope of the drag curve (1=Ae) is larger for the round tip in spite of the increased actual aspect ratio. The Oswald eciency factor for the wing with the square tip is By renormalizing the data for the round tip for actual projected planform area and the actual aspect ratio, the Oswald eciency factor for the wing with the round tip is Because of the possibility that higher loading can be maintained near the tip with the square end than near the tip with the round end at a given angle of attack, an elliptical load distribution can be approached more closely. Eect of Planform Shape The three elliptical planform wings have dierent span load distributions. Contributing factors are dierent spanwise ows in the boundary layer associated with dierent curvature of the quarterchord lines and dierent induced downwash distributions associated with each wing planform and wake shape. The eect of planform shape on the aerodynamic characteristics is presented in gure 20 for a Mach number of 0.5. The uncertainties in the lift and drag coecient measurements are smaller than the changes in the lift and drag coecients between wing A and wings B and C. Results from the curve ts in table 4 indicate that the angle of attack at zero lift becomes more negative by about 0:11 to 0:14 as the wingtip is swept aft. The drag coecient at zero lift does not change as the planform sweep changes. The eects of planform shape on the lift-curve slope and on the drag-curve slope (or wing eciency factor) will be discussed in the subsequent section on wing eciency. These results are typical of those 11